Question

Find the sum of the following series $1-\frac{n^2}{1^2}+\frac{n^2(n^2-1^2)}{1^2{.2}^2}-\frac{n^2(n^2-1^2)(n^2-2^2)}{1^2{.2}^2{.3}^2}+....$ to $n+1$ terms.

Answer 1

$\Rightarrow {(1+x)}^n=1+nx+\frac{n(n+1)}{2!}{x}^2+\frac{n(n-1)(n-2)}{3!}{x}^3+....$

${(1-\frac{1}{x})}^n=1-\frac{n}{x}+\frac{n(n+1)}{2!}\frac{1}{x^2}-\frac{n(n+1)(n+2)}{3!}\frac{1}{x^3}+.....$

$S=$ co efficient of $x^0$ in $\frac{{(x+1)}^n{(x-1)}^n}{x^n}$

$=\frac{{(x^2-1)}^n}{x^n}$ is coefficient of $x^n$

$={(-1)}^{n/2n}\frac{Cn}{2}$

$\therefore S=0$ if n is odd

and ${(-1)}^{n/2n}\frac{Cn}{2}$ if n is even